I'm really stuck on a question and i can't figure out how I should solve it.
I have a sum given by

Given the special case where

I have to prove that is logarithmically bounded by

And here's a *HINT* : Look at the terms

in

Well, I really don't get it...if someone could help me I would greatly appreciate!
ps. I'm glad I found thoses forums, I'll certainly become an active member, being a studient in computer sciences :P.
THANKS!

Jun 3rd 2013, 02:50 PM

emakarov

Re: Proof that a sum is bounded by given functions?

Prove the required claim by induction on m. This requires bounding from below and from above, as per the hint.

Jun 3rd 2013, 06:10 PM

Flexdec

Re: Proof that a sum is bounded by given functions?

Thank you for your reply!

I kinda guessed it was by induction but didn't think about the bounding from below. (I know... i had a rough day)
But still I'm confused about the induction hypothesis :S
Normally, inductions are all about a base case ( ) and an inductive step (do we need to show for ?)
I'm really confused! Thank you!

Jun 3rd 2013, 06:50 PM

Plato

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by Flexdec

Given the special case where
I have to prove that is logarithmically bounded by

Quote:

Originally Posted by Flexdec

I kinda guessed it was by induction but didn't think about the bounding from below. (I know... i had a rough day) But still I'm confused about the induction hypothesis :S
Normally, inductions are all about a base case ( ) and an inductive step (do we need to show for ?)

Actually the is very well know similar inequality:.
That gives you very quickly the RHS.

But I admit that I have not seen the LHS bound. So perhaps induction is the best approach.

Jun 4th 2013, 05:15 AM

emakarov

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by Flexdec

But still I'm confused about the induction hypothesis :S
Normally, inductions are all about a base case ( ) and an inductive step (do we need to show for ?)

Yes.

To be more explicit, prove P(m) for all by induction on m where P(m) is .

In the induction step, when you are proving P(m+1) from P(m), you know the lower and upper bounds on . Also, it is easy to get lower and upper bounds on by replacing the tail of the sum with its smallest or biggest term, respectively.

Jun 4th 2013, 06:45 AM

Flexdec

Re: Proof that a sum is bounded by given functions?

Ok so i give it a try. Given the special case , the sum is given by

Lets prove, by induction, the following logarithmic bounds

*edited ^

First, lets start with the base case

???? ????

well from there I get stuck, I think I did something wrong ...

Thank you for your help so far!!

Jun 4th 2013, 06:59 AM

emakarov

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by Flexdec

Given the special case , the sum is given by

Lets prove, by induction, the following logarithmic bounds

Why did you replace with ? The sum starts with 1.

Jun 4th 2013, 07:03 AM

Flexdec

Re: Proof that a sum is bounded by given functions?

I guess the hint messed with my head :P

So for base case it would be

right?

Jun 4th 2013, 07:12 AM

emakarov

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by Flexdec

So for base case it would be

right?

Yes.

Jun 4th 2013, 07:15 AM

Flexdec

Re: Proof that a sum is bounded by given functions?

and from there...

I'm lost :S

Jun 4th 2013, 10:46 AM

emakarov

Re: Proof that a sum is bounded by given functions?

Look again at post #5. You assume the induction hypothesis P(m), i.e., . From this assumption, you need to prove P(m+1), i.e., . Now, . The induction hypothesis gives you an upper and a lower bound on the first term of this sum, i.e., . As for the tail of , replace all terms with the smallest term to get a lower bound and with the largest term to get an upper bound. By adding the obtained lower bounds on and the tail you get a lower bound on , and by adding the obtained upper bounds on and the tail, you get an upper bound on . The two resulting inequalities are precisely P(m+1).

Jun 4th 2013, 01:01 PM

Flexdec

Re: Proof that a sum is bounded by given functions?

What do you mean by tail, all the terms except the first?
thanks!

Jun 4th 2013, 01:16 PM

emakarov

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by Flexdec

What do you mean by tail, all the terms except the first?

Yes, by the "tail of " I refer to .

Jun 4th 2013, 01:34 PM

Flexdec

Re: Proof that a sum is bounded by given functions?

And one last thing!!!

when you say replace all terms you mean replacing all the terms of the tail right?
I can't figure out how it can equal the lower and upper bounds when added to bounds.
For example, for lower bound:
where T is the modified tail you're talking about. would T be like
this is what I understood..

Jun 4th 2013, 02:04 PM

emakarov

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by Flexdec

when you say replace all terms you mean replacing all the terms of the tail right?

Yes, there are exactly in the tail of .

Quote:

Originally Posted by Flexdec

I can't figure out how it can equal the lower and upper bounds when added to bounds.
For example, for lower bound:
where T is the modified tail you're talking about. would T be like

You are correct. When we replace all terms with the smallest term , we get , and .

For the upper bound, we won't get the exact equality, but if we replace all terms with the largest term and add the upper bound of , i.e., m + 2, we get something that is ≤ (rather than =) the required upper bound m + 3.